# nLab multinomial coefficient

### Context

#### Combinatorics

combinatorics

enumerative combinatorics

graph theory

rewriting

### Polytopes

edit this sidebar

category: combinatorics

# Contents

## Idea

For $n \in \mathbb{N}_0$ a natural number or zero and $(k_i \in \mathbb{N}_0)_{i = 1}^r$ with $\underset{i}{\sum} k_i = k$, the corresponding multinomial coefficient

$\left( n \atop { k_1, k_2, \cdots, k_r } \right) \;\coloneqq\; \frac{ n! }{ k_1 ! \, k_2 ! \, \cdots \, k_r } \;\in\; \mathbb{N}$

is the quotient of the factorial of $n$ by the multiplication of the factorials of the $k_i$.

This number is the number of ways of drawing $k_1$ elements and then $k_2$ elements and so forth from $n$ elements, all in an unordered way.

For $r = 2$ this is the binomial coefficient.